All right. I am now inclined to switch topics a little (I am looking forward to saying a few words about local cohomology), so I will sketch a few details in the present post. The goal is to compute the sheaf cohomology groups of the canonical line bundles on projective space. The argument will follow EGA III.2; Hartshorne does essentially the same thing (namely, analysis of the Cech complex) but without the Koszul machinery, so his approach seems more opaque to me.
Now, let us compute the cohomology of projective space over a ring . Note that is quasi-compact and separated, so we can compute the Cech cohomology by the above machinery. That is, we will use the Koszul-Cech connection discussed two days ago. In particular, we will consider the quasi-coherent sheaf
(So here , to use the notation of two days ago.) We have the sections for . It is a basic fact that the sets are affine subsets of ; in fact they are sometimes called basic open affines. Each is isomorphic to .
In order to apply the machinery, we will need to first find .
Lemma 45 The global sections of are precisely the degree polynomials. (When , there are none.)
Proof: is the sheaf associated to the graded -module whose graded part is defined by . The sections over are isomorphic to . But is the intersection of the localizations,
This identity makes sense as graded rings. Thus if something lies in for all , then it must lie in , and is obviously of degree . In particular, the lemma tells us that
as, in fact, a ring!
Proposition 46 The Cech cohomology is as follows. It is zero if . For , it is isomorphic as a graded -module to . For , it is a free -module on rational functions where .
Proof: Indeed, we know that Cech cohomology is a direct limit of Koszul cohomologies. In particular, we have for ,
where . However, the sequence , and consequently all powers , are regular sequences on of length . This implies that in Koszul cohomology, by a lemma below, that
for . This implies the vanishing of for .
Last, but not least, we have to do . Here again we can use Koszul cohomology. We know that this is the direct limit . As we will see below, this is the direct limit of the over getting larger, where the maps are multiplication by .
To compute the colimit of this, note that at any element in the colimit is measured by “its distance from the end.” In particular, the elements are spanned by the negative monomials for . For each large, this negative monomial corresponds to the element
in a way that is manifestly compatible with the maps in the colimit diagram.
Theorem 47 Let be a ring. The cohomology of the line bundle on projective space is as follows. If , then is the collection of polynomials of degree if , and otherwise. If , then . If , then is zero for , but for other is free on the set of negative monomials for .
Proof: This now follows from the previous result, if we split the gradings. Note that we are using the fact that derived functor cohomology coincides with Cech cohomology for an affine covering on a separated scheme.
It is an interesting and important fact that all the cohomology groups here are finitely generated, which is a special property of projective space. It is also a curious fact that if is any line bundle of this form, then for , we have
These will figure in the future.
2.7. A loose end: self-duality of the Koszul complex
Earlier, we saw that if is an -sequence, then the complex is acyclic in dimension not zero. But in the above computation, we used a dual analog:
Proposition 48 Let be an -sequence. Then the complex is acyclic in dimension not .
Proof: This will now follow from a dualization of the argument. Namely, the claim is that the Koszul chain complexes and cochain complexes are dual to each other. That is, is the cochain complex with the degrees reversed. In fact, let us state this more carefully.
Let be a free module of rank . Let . Then is isomorphic to . In fact, if is a basis for , then the map can be taken to be
where the are the complementary set to the and the are ordered appropriately. Anyway, the point is that we have an isomorphism
One can show that the differential on is the transpose of the differential on . This is what was said about self-duality.
Corollary 49 Let be an -module and be a sequence. Then .
This is now clear from the computation of and the self-duality.